cp's OEIS Frontend

A356637 a(n) = A000265(A263931(n)).

%I A356637 #29 Sep 08 2022 05:47:33
%S A356637 1,1,1,1,1,9,3,3,45,5,1,21,7,175,675,45,45,1485,5775,5775,45045,2145,
%T A356637 195,8775,2925,5733,22491,833,6545,373065,24871,24871,1566873,3086265,
%U A356637 181545,357903,39767,39767,156975,309925,61985,5020785,239085,20322225,160730325
%N A356637 a(n) = A000265(A263931(n)).
%C A356637 Let n >= 5. If a(n) is squarefree, then 2 divides binomial(2*n, n) more than once and is the only prime that does so. There is only a finite number of such cases (see A059097).
%C A356637 An efficient algorithm for the calculation is available, which is based on prime factorization. See the SageMath implementation. The main application is the efficient calculation of the central binomial coefficient, which is the product of this sequence, the Glaisher/Gould sequence, and the upper primorial function (see the formula section).
%C A356637 Since the central binomial coefficient is a bisection of the swinging factorial A056040, and the swinging factorial, in turn, is the building block for an efficient algorithm for the computation of the factorial function, the terms of this sequence occur as factors in all these computations. See the links for details.
%H A356637 Peter Luschny, <a href="/A000142/a000142.pdf">Swing, divide and conquer the factorial</a>, (excerpt).
%H A356637 Peter Luschny, <a href="https://github.com/PeterLuschny/Fast-Factorial-Functions">Fast Factorial Functions</a>, a code repository.
%H A356637 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html">Erdős Squarefree Conjecture</a>.
%F A356637 A000984(n) = a(n) * A001316(n) * A261130(n) for n >= 2.
%e A356637 Let n = 22 and consider the prime factorization of m = binomial(2*n, n):
%e A356637 2^3 * [3 * 5 * 13] * 23 * 29 * 31 * 37 * 41 * 43. Then a(22) = 3 * 5 * 13. This is what is left after the 'prime tail' A261130(n) and the 'prime head' A006519(m) = A001316(n) have been cut off.
%p A356637 A263931 := n -> binomial(2*n, n) / convert(select(isprime, {$n+1..2*n}), `*`):
%p A356637 A000265 := n -> n / 2^padic[ordp](n, 2):
%p A356637 seq(A000265(A263931(n)), n = 0..45);
%o A356637 (SageMath)
%o A356637 def A356637(n: int) -> int:
%o A356637     m = 2 * n
%o A356637     if m < 5: return 1
%o A356637     sqrtm = isqrt(m) + 1
%o A356637     R = prime_range(sqrtm, m // 3 + 1)
%o A356637     factors = [x for x in R if is_odd(m // x)]
%o A356637     for prime in prime_range(3, sqrtm):
%o A356637         p: int = 1
%o A356637         q: int = m
%o A356637         while True:
%o A356637             q //= prime
%o A356637             if q == 0:
%o A356637                 break
%o A356637             if q & 1 == 1:
%o A356637                 p *= prime
%o A356637         if p > 1:
%o A356637             factors.append(p)
%o A356637     return product(factors)
%o A356637 print([A356637(n) for n in range(45)])
%Y A356637 Cf. A000265, A263931, A261130, A006519, A000984, A001316, A059097, A056040.
%K A356637 nonn
%O A356637 0,6
%A A356637 _Peter Luschny_, Sep 07 2022